A model of the motion of particles in a quantum harmonic oscillator on $ℝ$ was proposed ( Kowalski, 2022 ). This is a skew-product transformation ${\hat{S}}_p$ where $p\in \left(\frac{1}{2},\frac{1}{\sqrt[3]{2}}\right).$ The aim of the work is to show that ${\hat{S}}_p$ has the Bernoulli property. Therefore, we will prove the ergodicity of ${\hat{S}}_p$. We consider its isomorphic version $S_p$ which is a random walk on $I=\left[0,1\right]$. Firstly, we will describe the probability space on which $S_p$ operates. It is a product space $\left(\Omega \times I,ℬ\times \mathcal{A},{\mu }_p\times {\mu }_F\right)$ where $\Omega$ is the space ${\left\{0,1\right\}}^{\mathbb{N}}$, $\mathbb{N}=\left\{0,1,2,\cdots \right\}$, with the $\left(p,1-p\right)$-Bernoulli measure ${\mu }_p$ on $\left(\Omega ,ℬ\right)$. Here $ℬ$ is the Borel product $\sigma$-algebra and $\mathcal{A}$ the Borel $\sigma$-algebra of subsets of $ℝ$. The measure ${\mu }_F$ has a probability distribution $F$ which is given by the mathematical description of the oscillator in ( Kowalski, 2022 ) as follows. One-dimensional quantum harmonic oscillator $\Psi$ satisfies the Schrödinger equation

$i\hslash \frac{\partial \Psi }{\partial t}=H\Psi$

where $H=\frac{1}{2}\left(P^2+Q^2\right)$ is the Hamiltonian. Here $P=-i\frac{\text{d}}{\text{d}x}$ and $Q$ is multi-plication by $x$. Let us consider the $n$-quantum state solution

${\Psi }_n\left(x,t\right)=\frac{1}{\sqrt{n!2^n\sqrt{\pi }}}{H}_n\left(x\right)\exp \left(-\frac{i}{2\hslash }\left(2n+1\right)t\right)\exp \left(-\frac{x^2}{2}\right).$

for $n\ge 0$. Here $H_n$ is $n$th Hermite polynomial i.e.

$H_n\left(x\right)={\left(-1\right)}^n{\text{e}}^{x^2}\frac{{\text{d}}^n}{\text{d}{x}^n}{\text{e}}^{-x^2}.$

The quantum interpretation of

${\Phi }_n\left(x\right)={\displaystyle {\int }_{-\infty }^x{\left|{\Psi }_n\left(y,t\right)\right|}^2\text{d}y}=\frac{1}{n!2^n\sqrt{\pi }}{\displaystyle {\int }_{-\infty }^x{H}_n^2\left(y\right)\exp \left(-y^2\right)\text{d}y}$

is the probability distribution of occurrence of a particle in $ℝ$. In Theorem 4.1 ( Kowalski, 2017 ), it has been observed that for $n\ge 1$ there is a unique partition on intervals $\left\{I_k:k=1,\cdots ,2n\right\}$ of $ℝ$ such that ${\mu }_n\left(I_k\right)={\mu }_{n-1}\left(I_k\right)$ where the measure ${\mu }_n$ has distribution ${\Phi }_n$. The endpoints of intervals come from the equivalence

${\Phi }_n\left(x\right)={\Phi }_{n-1}\left(x\right)\iff H_n\left(x\right)H_{n-1}\left(x\right)=0.$

It is convenient to consider the unit interval instead of $ℝ$, therefore, we use the map ${\Phi }_{n-1}:ℝ\to \left(0,1\right)$. ${\Phi }_{n-1}$ arise as the distribution function of the Lebesgue measure on the unit interval and ${\Phi }_n$ as $F^{\left(n\right)}={\Phi }_n\circ {\Phi }_{n-1}^{-1}$. Here ${\Phi }_{n-1}^{-1}$ is the inverse function of ${\Phi }_{n-1}$. We will denote $F^{\left(n\right)}$ by $F$ as $n$ is fixed. Next, we will describe the construction of the skew-product $S_p$. Let $\sigma$ be the one-sided shift on $\Omega$ i.e. $\sigma \left(\omega \right)\left(i\right)=\omega \left(i+1\right)$. Let us assume that conditions of Remark 2.6 ( Kowalski, 2022 ) hold for some $p\in \left(\frac{1}{2},\frac{1}{\sqrt[3]{2}}\right]$ and for some $1\le k\le 2n-1$. Then we get the step skew product transformation in the space $\Omega \times J_{k+1}$ as follows

$S_p\left(\omega ,u\right)=\{\begin{array}{ll}\left(\sigma \left(\omega \right),g^{-1}\left(u\right)\right)\hfill & \text{for}{\omega }_0=1\text{and}\hfill \\ \left(\sigma \left(\omega \right),{\left(2u-g\left(u\right)\right)}^{-1}\right)\hfill & \text{for}{\omega }_0=0.\hfill \end{array}$

Here $g\left(u\right)$ and $2u-g\left(u\right)$ are the increasing self-homeomorphisms of $J_{k+1}$. The skew product as above preserves the measure ${\mu }_p\times {\mu }_F$ on $\Omega \times J_{k+1}$. Here $J_{k+1}={\Phi }_{n-1}\left(I_{k+1}\right)$ for $k=1,\cdots ,2n-1$. The formula ${\left(2u-g\left(u\right)\right)}^{-1}$ for the second homeomorphism is equivalent to $S_p$ invariance of the measure ${\mu }_{\frac{1}{2}}\times \Lambda$ where $\Lambda$ denotes the Lebesgue measure. Section 3 is dedicated to the binomial model for asset prices.

Firstly, we normalize the ${\mu }_F$ measure on $J_{k+1}$ and denote it by ${\bar{\mu }}_F$.

Definition 2.1. $\left(S_p,{\mu }_p\times {\bar{\mu }}_F\right)$ has the Bernoulli property if

${\bar{S}}_p\left(\bar{\omega },u\right)=\{\begin{array}{ll}\left(\bar{\sigma }\left(\bar{\omega }\right),g^{-1}\left(u\right)\right)\hfill & \text{for}{\bar{\omega }}_0=1\text{and}\hfill \\ \left(\bar{\sigma }\left(\bar{\omega }\right),{\left(2u-g\left(u\right)\right)}^{-1}\right)\hfill & \text{for}{\bar{\omega }}_0=0.\hfill \end{array}$

is a Bernoulli automorphism.

Here $\bar{\omega }\in {\left\{0,1\right\}}^{\mathcal{Z}}$ and $\bar{\sigma }$ is the two-sided $\left(p,1-p\right)$-Bernoulli shift.

Theorem 2.2. The skew-product transformation $S_p$ has the Bernoulli property.

Proof. By ergodic decomposition of ${\mu }_p\times {\bar{\mu }}_F$ ( Kifer, 1986: p. 193 , Theorem 1.1) there exists ergodic $S_p$-invariant measure ${\mu }_p\times {\mu }_G$ such that ${\mu }_G\left(\left\{b_k\right\}\right)={\mu }_G\left(\left\{b_{k+1}\right\}\right)=0$. Hence ${\mu }_G$ has the dense support property by Lemma 3 ( Kowalski, 2009 ), i.e., if ${\mu }_G\left(A\right)={\mu }_G\left(J_{k+1}\right)$ then $\bar{A}=J_{k+1}$. Therefore

$M_p\left(S_p\right)=\text{conv}\left\{{\mu }_p\times {\delta }_{\left\{b_k\right\}},{\mu }_p\times {\delta }_{\left\{b_{k+1}\right\}},{\mu }_p\times {\mu }_G\right\}$

by Theorem 1 ( Kowalski, 2003 ). Here $M_p\left(S_p\right)$ denotes the set of $S_p$-invariant probability measures $\mu$ on $\Omega \times J_{k+1}$ such that the left marginal measure of $\mu$ is ${\mu }_p.$ So we conclude that ${\bar{\mu }}_F={\mu }_G$ as ${\mu }_p\times {\bar{\mu }}_F\in M_p\left(S_p\right)$ and ${\bar{\mu }}_F\left(\left\{b_k\right\}\right)={\bar{\mu }}_F\left(\left\{b_{k+1}\right\}\right)=0$. Now we are able to use Corollary 5.2 ( Kowalski, 2019 ) to conclude that $\left(S_p,{\mu }_p\times {\bar{\mu }}_F\right)$ has the Bernoulli property. ☐

We define the skew product transformation ${\hat{S}}_p\left(\omega ,x\right)$ in the space $\Omega \times I_{k+1}$ by putting $H_g$ and $H_{2u-g\left(u\right)}$ instead of $g$ and $2u-g\left(u\right)$ in the definition of $S_p\left(\omega ,u\right)$. Here

$H_g\left(x\right)={\Phi }_{n-1}^{-1}\left(g\left({\Phi }_{n-1}\left(x\right)\right)\right)\text{for}x\in I_{k+1}.$

This skew product preserves the measure ${\mu }_p\times {\mu }_{{\Phi }_n}$ on $\Omega \times I_{k+1}$ where ${\mu }_{{\Phi }_n}$ has distribution ${\Phi }_n$. Moreover, its natural extension to automorphism is Bernoulli one. For a detailed description, see ( Kowalski, 2022 ).

A generalized one-asset binomial model, see Bahsoun et al. (2007) , Kowalski, (2017) , is a random walk on I. Here $x\in I$ represents the security price. Physical phenomena at the quantum level, where randomness rules, can well describe the macro world. Therefore, we can take the skew-product ${\hat{S}}_p$ as a $\left(n-1\right)$-asset binomal model. For $n=3$, see ( Kowalski, 2022 , Figure 4), we have

$S_d\left(x\right)=h_1\left(x\right)\text{and}{S}_u\left(x\right)=h_0\left(x\right)\text{for}x\in I_4$

where $I_4=\left[0,\frac{\sqrt{2}}{2}\right]$. This is the price of one share of common stock of a particular corporation. The subscript $u$ of $S_u$ illustrates that the transformation $S_u$ contains a law that moves the price up, and the subscript $d$ of $S_d$ illustrates that $S_d$ contains a law that moves the price down. The Bernoulli property of ${\hat{S}}_p\left(\omega ,x\right)$ implies the mixing of ${\hat{S}}_p$. Hence

$\underset{m\to \infty }{\lim }{\displaystyle \int 1_{\Omega \times A}\left({\hat{S}}_p{}^m\right)f\text{d}{\mu }_p\times {\mu }_{{\Phi }_n}}={\mu }_{{\Phi }_n}\left(A\right){\displaystyle \int f\text{d}{\mu }_{{\Phi }_n}}$

for every $A\in \mathcal{A}$ and $f\in L_{\infty }\left({\mu }_{{\Phi }_n}\right)$. Therefore

$\underset{m\to \infty }{\lim }{\mu }_p\left(\left\{\omega :{\hat{S}}_p^m\left(\omega ,x\right)\in \Omega \times A\right\}\right)={\mu }_{{\Phi }_n}\left(A\right)$

in weak- $L_1\left({\mu }_{{\text{Φ}}_n}\right)$ convergence.

In a similar way, we consider the second financial asset, price-adjacent to the previous one. Here we take

$I_5=\left[\frac{\sqrt{2}}{2}.\frac{\sqrt{6}}{2}\right]$ and

$S_d\left(x\right)=h_0\left(x\right)\text{and}{S}_u\left(x\right)=h_1\left(x\right)\text{for}x\in I_5.$

We can turn the interval $\left[0,\frac{\sqrt{6}}{2}\right]$ into the interval $\left[0,1\right]$ by linearly changing the variables.

For example, the above may be illustrated by the continuous quotations exchange of shares of a company listed on the Warsaw Stock Exchange on May 7, 2024. This is Atende S.A. Here for $n=20$ there are two adjacent zeros of the Hermite polynomials $H_n$ and $H_{n-1}$ such that $x_1\approx 2.79$, $x_2\approx 3,16$. If we ignore prices at which the turnover is less than 100, the firm’s price distribution strictly increases as does ${\Phi }_{20}$. Moreover, $p\approx 0.5$, so it is a state of equilibrium.